We denote by \(s_J^B\)  and \(s_I^B\)  the Bismut-Ricci scalar curvatures of \((g, J)\)  and \((g, I)\) , respectively. By Proposition REF  and [1]}, we know that
\( \rho ^B_J- \rho ^B_I = \frac{1}{2}{\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \qquad s_J^B - s_I^B = -\delta ^g(\theta _J - \theta _I). \) 
Using that \(F=-2g(I+J)^{-1}\)  and the above formula, we compute
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= s_J^B - \big \langle \rho ^B_J, \omega _J + \omega _I\big \rangle _g = \frac{1}{2} s_J^B - \big \langle \rho ^B_J, \omega _I\big \rangle _g \\&= \frac{1}{2} (s_J^B - s_I^B) - \frac{1}{2}\Big \langle {\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \omega _I\Big \rangle _g\\=&\ -\frac{1}{2}\delta ^g(\theta _J - \theta _I)- \frac{1}{2}\Big \langle {\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F, \omega _I\Big \rangle _g.\end{split} \) 
We next use that, by (REF ),
\({\mathcal {L}}_{(\theta _J^{\sharp }-\theta _{I}^{\sharp })} F = {\mathrmsl d}\left(F(\theta _J^{\sharp }-\theta _I^{\sharp })\right) = {\mathrmsl d}\left(I(\theta _J - \theta _I) + Ib(\theta _J^{\sharp }-\theta _I^{\sharp })\right),\) 
so we continue the computation
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= -\frac{1}{2}\delta ^g(\theta _J - \theta _I)- \frac{1}{2}\Big \langle {\mathrmsl d}\left(I\left( (\theta _J - \theta _I) + b(\theta _J^{\sharp }-\theta _I^{\sharp })\right)\right), \omega _I\Big \rangle _g \\&= \frac{1}{2}\left(\delta ^g\left(b(\theta _J^{\sharp }-\theta _I^{\sharp })\right) + \big \langle \theta _J - \theta _I + b(\theta _J^{\sharp }-\theta _I^{\sharp }), \theta _I\big \rangle _g\right),\end{split}\) 
where we have used the identity
\(\langle {\mathrmsl d}I \alpha , \omega _I\rangle _g= -\delta ^g \alpha - \langle \alpha , \theta _I\rangle _g.\) 
We next use the identity (which follows from Lemma REF )
\( {\mathrmsl d}\log \det (I+J) = \theta _I + \theta _J + b(\theta _J^{\sharp }-\theta _I^{\sharp })\) 
in order to express the terms containing \(b(\theta _J^{\sharp }-\theta _I^{\sharp })\) . We thus get
\(\begin{split}2\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] &= \frac{1}{2}\left(-\Delta ^g \log \det (I+J) - \delta ^g(\theta _I +\theta _J) + \big \langle - 2\theta _I + {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g\right) \\&= \frac{1}{2}\left(-\Delta ^g \log \det (I+J) - \delta ^g(\theta _I +\theta _J) - 2|\theta _I|^2_g + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g\right).\end{split}\) 
We finally use that (as \(F= -2g(I+J)^{-1}\) )
\(F^{[m]} = 2^m \det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \) 
and integration by parts to get
\(\begin{split}&\int _M\left[\operatorname{tr}_{\omega _J}(\rho ^B_J) - \operatorname{tr}_{F}(\rho ^B_J)\right] F^{[m]} \\&= 2^{m-2}\int _M\left(-\Delta ^g \log \det (I+J) + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g - \delta ^g(\theta _I +\theta _J) - 2|\theta _I|^2_g \right)\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= 2^{m-2}\int _M\left(-\Delta ^g \log \det (I+J) + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I\big \rangle _g - |\theta _I|^2_g-|\theta _J|^2_g-2\delta ^g\theta _J \right)\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= 2^{m-2}\int _M\left[-\frac{1}{2}\Big |{\mathrmsl d}\log \det (I+J)\Big |^2_g + \big \langle {\mathrmsl d}\log \det (I+J), \theta _I+\theta _J\big \rangle _g-|\theta _I|^2_g-|\theta _J|^2_g \right]\det (I+J)^{-\frac{1}{2}}\, \omega _J^{[m]} \\&= -\frac{1}{8}\int _M\left[ \Big |{\mathrmsl d}\log \det (I+J)-\theta _I-\theta _J\Big |^2_g+|\theta _I-\theta _J|^2_g \right]F^{[m]}\end{split}\) 
where we have used the relation \(\delta ^g \theta _I + |\theta _I|^2_g =|H|_g^2=\delta ^g\theta _J+|\theta _J|^2_g\)  from [1]}.
